On the other hand, it was observed by Sogge [1]} that considering space-time Lebesgue norms of the solution to the Euclidean wave equation yields a gain of regularity over the fixed-time estimates in (REF ). More precisely, in [1]} Sogge formulated the local smoothing conjecture for the Euclidean wave equation, which states that
\(\Vert  e^{it \sqrt{- \Delta }} f \Vert _{L_t^p([0,1],L^p(\mathbb {R}^d))} \lesssim _{\varepsilon } \Vert  f \Vert _{W^{\sigma (p)+\varepsilon ,p}(\mathbb {R}^d)}\) 
